1888.] Bending and Vibration of thin Elastic Shells. 107 



the general condition necessary to be satisfied at a free edge is in 

 fact violated by such a deformation as (1). But the condition in 

 question* contains terms proportional to the first and to the third 

 powers respectively of the thickness, the coefficients of the former 

 involving as factors the extensions and shear of the middle surface. 

 It appears to me that when the thickness is diminished without limit, 

 the fulfilment of the boundary condition requires only that the middle 

 surface be unstretched, precisely the requirement satisfied by solutions 

 such as (1). 



Of course, so long as the thickness is finite, the forces in operation 

 will entail some stretching of the middle surface, and the amount of 

 this stretching will depend on circumstances. A good example is 

 afforded by a circular cylinder with plane edges perpendicular to the 

 axis. Let normal forces locally applied at the extremities of one 

 diameter of the central section cause a given shortening of that 

 diameter. That the potential energy may be a minimum, the de- 

 formation mast assume more and more the character of mere bending 

 as the thickness is reduced. The only kind of bending that can occur 

 in this case is the purely cylindrical one in which every normal 

 section is similarly deformed, and then the potential energy is propor- 

 tional to the total length of the cylinder. We see, therefore, that if 

 the cylinder be very long, the energy of bending corresponding to the 

 given local contraction of the central diameter may become very 

 great, and a heavy strain is thrown upon the principle that the 

 deformation of minimum energy is one of pure bending. 



If the small thickness of the shell be regarded as given, a point 

 will at last be attained when the energy can be made least by a 

 sensible local stretching of the middle surface such as will dispense 

 with the uniform bending otherwise necessary over so great a length. 

 But even in this extreme case it seems correct to say that, when the 

 thickness is sufficiently reduced, the deformation tends to become one 

 of pure bending. 



At first sight it may appear strange that of two terms in an ex- 

 pression of the potential energy, the one proportional to the cube of 

 the thickness is to be retained, while that proportional to the first 

 power may be omitted. The fact, however, is that the large potential 

 enero-y which would accompany any stretching of the middle surface 

 is the very reason why such stretching will not occur. The compara- 

 tive largeness of the coefficient (proportional to the first power of the 

 thickness) is more than neutralised by the smallness of the stretching 

 itself, to the square of which the energy is proportional. 



In general, if y/ 1 be the coordinate measuring the violation of the 

 tie which is supposed to be more and more insisted upon by increasing 



* See bis equation (33). 



